CamTutor

Question - Solving a System of Linear Equations Using Substitution or Elimination

math algebra system of linear equations simultaneous equations substitution elimination

Solution:

The system of equations is:

\[ \begin{align*} 5x + 4y &= 58 \quad \text{(1)} \\ 3x + 7y &= 67 \quad \text{(2)} \end{align*} \]

To solve the system, we can use either substitution or elimination. Here, we will use the elimination method. The goal is to eliminate one variable by making the coefficients of either \( x \) or \( y \) the same in both equations.

Multiply equation (1) by 3 and equation (2) by 5:

\[ \begin{align*} 3(5x + 4y) &= 3(58) \quad \Rightarrow \quad 15x + 12y &= 174 \quad \text{(3)} \\ 5(3x + 7y) &= 5(67) \quad \Rightarrow \quad 15x + 35y &= 335 \quad \text{(4)} \end{align*} \]

Subtract equation (3) from equation (4):

\[ \begin{align*} (15x + 35y) - (15x + 12y) &= 335 - 174 \\ 15x + 35y - 15x - 12y &= 161 \\ 23y &= 161 \end{align*} \]

Divide by 23 to find \( y \):

\[ \begin{align*} y &= \frac{161}{23} \\ y &= 7 \end{align*} \]

Substitute \( y = 7 \) into equation (1):

\[ \begin{align*} 5x + 4(7) &= 58 \\ 5x + 28 &= 58 \\ 5x &= 58 - 28 \\ 5x &= 30 \end{align*} \]

Divide by 5 to find \( x \):

\[ \begin{align*} x &= \frac{30}{5} \\ x &= 6 \end{align*} \]

The solution set is \( (x, y) = (6, 7) \).